Constructing 7-Clusters

A set of n-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an n-cluster (in R 2 ). We determine the smallest existent 7-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 dierent 7-clusters, some of them having huge integer edge lengths. On the way, we exhaustively determined all Heronian triangles with largest edge length up to 6 10 6 .

[1]  R. Guy.  Unsolved Problems in Number Theory , 1981 .

[2]  Vladimir I. Benediktovich.  On rational approximation of a geometric graph , 2013, Discret. Math..

[3]  S. Ulam.  A collection of mathematical problems , 1960 .

[4]  József Solymosi,et al.  On a Question of Erdős and Ulam , 2008, Discret. Comput. Geom..

[5]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[6]  Sascha Kurz.  On the characteristic of integral point sets in 𝔼m , 2006, Australas. J Comb..

[7]  J.-A. Serret.  Sur un théorème relatif aux nombres entiers. , 1848 .

[9]  Landon Curt Noll,et al.  $n$-clusters for $1 , 1989 .

[10]  Reinhard Laue,et al.  Bounds for the minimum diameter of integral point sets , 2007, Australas. J Comb..

[11]  Alexander R. Perlis,et al.  Heronian Tetrahedra Are Lattice Tetrahedra , 2013, Am. Math. Mon..

[12]  Sascha Kurz.  Konstruktion und Eigenschaften ganzzahliger Punktmengen , 2005 .

[13]  R. Read.  Every one a Winner or how to Avoid Isomorphism Search when Cataloguing Combinatorial Configurations , 1978 .

[14]  John Brillhart,et al.  Note on Representing a Prime as a Sum of Two Squares , 1972 .

[15]  József Solymosi.  Note on Integral Distances , 2003, Discret. Comput. Geom..

[16]  Sascha Kurz.  On the generation of Heronian triangles , 2014, 1401.6150.

[17]  T. Broadbent.  Diophantine Analysis , 1958, Nature.

[18]  Heron Triangles with Two Fixed Sides , 2006, math/0608185.

[19]  Alfred Wassermann,et al.  On the minimum diameter of plane integral point sets , 2011, Ars Comb..

[20]  Sascha Kurz,et al.  There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle , 2008, Discret. Comput. Geom..

[21]  Rational and Heron tetrahedra , 2006 .

[22]  James F. Geelen,et al.  Straight line embeddings of cubic planar graphs with integer edge lengths , 2008, J. Graph Theory.

[23]  Paul Yiu.  Heronian Triangles Are Lattice Triangles , 2001, Am. Math. Mon..

[24]  Jan Fricke.  On Heron Simplices and Integer Embedding , 2008 .

[25]  K. Menger.  Untersuchungen über allgemeine Metrik , 1928 .